Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [126,12,Mod(37,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.37"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 126.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-128,0,-4096,1420] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(96.8112407505\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} + 131491 x^{6} + 44838722 x^{5} + 18152831051 x^{4} + 2926931386118 x^{3} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{3}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 37.2
Root \(-110.360 - 191.150i\) of defining polynomial
Character \(\chi\) \(=\) 126.37
Dual form 126.12.g.d.109.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-16.0000 + 27.7128i) q^{2} +(-512.000 - 886.810i) q^{4} +(1017.54 - 1762.43i) q^{5} +(42078.2 - 14378.9i) q^{7} +32768.0 q^{8} +(32561.3 + 56397.9i) q^{10} +(-39381.2 - 68210.2i) q^{11} -788558. q^{13} +(-274771. + 1.39617e6i) q^{14} +(-524288. + 908093. i) q^{16} +(-5.11629e6 - 8.86167e6i) q^{17} +(1.03302e7 - 1.78924e7i) q^{19} -2.08392e6 q^{20} +2.52039e6 q^{22} +(3.67657e6 - 6.36801e6i) q^{23} +(2.23433e7 + 3.86997e7i) q^{25} +(1.26169e7 - 2.18532e7i) q^{26} +(-3.42954e7 - 2.99534e7i) q^{28} -9.93024e7 q^{29} +(1.00917e8 + 1.74793e8i) q^{31} +(-1.67772e7 - 2.90590e7i) q^{32} +3.27443e8 q^{34} +(1.74744e7 - 8.87911e7i) q^{35} +(1.95399e8 - 3.38442e8i) q^{37} +(3.30566e8 + 5.72557e8i) q^{38} +(3.33428e7 - 5.77514e7i) q^{40} -7.80704e8 q^{41} +2.17607e7 q^{43} +(-4.03263e7 + 6.98472e7i) q^{44} +(1.17650e8 + 2.03776e8i) q^{46} +(-1.51556e9 + 2.62504e9i) q^{47} +(1.56382e9 - 1.21008e9i) q^{49} -1.42997e9 q^{50} +(4.03742e8 + 6.99301e8i) q^{52} +(-1.98519e9 - 3.43845e9i) q^{53} -1.60288e8 q^{55} +(1.37882e9 - 4.71168e8i) q^{56} +(1.58884e9 - 2.75195e9i) q^{58} +(1.37766e9 + 2.38619e9i) q^{59} +(-5.25879e9 + 9.10849e9i) q^{61} -6.45867e9 q^{62} +1.07374e9 q^{64} +(-8.02391e8 + 1.38978e9i) q^{65} +(1.14527e9 + 1.98367e9i) q^{67} +(-5.23908e9 + 9.07435e9i) q^{68} +(2.18106e9 + 1.90492e9i) q^{70} -4.54366e9 q^{71} +(-6.98203e9 - 1.20932e10i) q^{73} +(6.25278e9 + 1.08301e10i) q^{74} -2.11562e10 q^{76} +(-2.63788e9 - 2.30390e9i) q^{77} +(-1.39090e10 + 2.40910e10i) q^{79} +(1.06697e9 + 1.84805e9i) q^{80} +(1.24913e10 - 2.16355e10i) q^{82} +6.17383e10 q^{83} -2.08241e10 q^{85} +(-3.48172e8 + 6.03052e8i) q^{86} +(-1.29044e9 - 2.23511e9i) q^{88} +(1.20759e10 - 2.09160e10i) q^{89} +(-3.31811e10 + 1.13386e10i) q^{91} -7.52962e9 q^{92} +(-4.84981e10 - 8.40011e10i) q^{94} +(-2.10228e10 - 3.64125e10i) q^{95} -1.03350e11 q^{97} +(8.51348e9 + 6.26991e10i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 128 q^{2} - 4096 q^{4} + 1420 q^{5} - 66362 q^{7} + 262144 q^{8} + 45440 q^{10} + 861962 q^{11} + 836748 q^{13} + 1361216 q^{14} - 4194304 q^{16} - 5647852 q^{17} + 16668730 q^{19} - 2908160 q^{20}+ \cdots + 80526496576 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.0000 + 27.7128i −0.353553 + 0.612372i
\(3\) 0 0
\(4\) −512.000 − 886.810i −0.250000 − 0.433013i
\(5\) 1017.54 − 1762.43i 0.145619 − 0.252219i −0.783985 − 0.620780i \(-0.786816\pi\)
0.929604 + 0.368561i \(0.120149\pi\)
\(6\) 0 0
\(7\) 42078.2 − 14378.9i 0.946276 − 0.323360i
\(8\) 32768.0 0.353553
\(9\) 0 0
\(10\) 32561.3 + 56397.9i 0.102968 + 0.178346i
\(11\) −39381.2 − 68210.2i −0.0737274 − 0.127700i 0.826805 − 0.562489i \(-0.190156\pi\)
−0.900532 + 0.434789i \(0.856823\pi\)
\(12\) 0 0
\(13\) −788558. −0.589041 −0.294520 − 0.955645i \(-0.595160\pi\)
−0.294520 + 0.955645i \(0.595160\pi\)
\(14\) −274771. + 1.39617e6i −0.136542 + 0.693798i
\(15\) 0 0
\(16\) −524288. + 908093.i −0.125000 + 0.216506i
\(17\) −5.11629e6 − 8.86167e6i −0.873949 − 1.51372i −0.857878 − 0.513853i \(-0.828218\pi\)
−0.0160707 − 0.999871i \(-0.505116\pi\)
\(18\) 0 0
\(19\) 1.03302e7 − 1.78924e7i 0.957113 − 1.65777i 0.227656 − 0.973742i \(-0.426894\pi\)
0.729457 − 0.684027i \(-0.239773\pi\)
\(20\) −2.08392e6 −0.145619
\(21\) 0 0
\(22\) 2.52039e6 0.104266
\(23\) 3.67657e6 − 6.36801e6i 0.119108 − 0.206301i −0.800307 − 0.599591i \(-0.795330\pi\)
0.919414 + 0.393290i \(0.128663\pi\)
\(24\) 0 0
\(25\) 2.23433e7 + 3.86997e7i 0.457590 + 0.792570i
\(26\) 1.26169e7 − 2.18532e7i 0.208257 − 0.360712i
\(27\) 0 0
\(28\) −3.42954e7 − 2.99534e7i −0.376588 − 0.328909i
\(29\) −9.93024e7 −0.899023 −0.449511 − 0.893275i \(-0.648402\pi\)
−0.449511 + 0.893275i \(0.648402\pi\)
\(30\) 0 0
\(31\) 1.00917e8 + 1.74793e8i 0.633102 + 1.09657i 0.986914 + 0.161249i \(0.0515521\pi\)
−0.353811 + 0.935317i \(0.615115\pi\)
\(32\) −1.67772e7 − 2.90590e7i −0.0883883 − 0.153093i
\(33\) 0 0
\(34\) 3.27443e8 1.23595
\(35\) 1.74744e7 − 8.87911e7i 0.0562379 − 0.285756i
\(36\) 0 0
\(37\) 1.95399e8 − 3.38442e8i 0.463248 − 0.802370i −0.535872 − 0.844299i \(-0.680017\pi\)
0.999121 + 0.0419295i \(0.0133505\pi\)
\(38\) 3.30566e8 + 5.72557e8i 0.676781 + 1.17222i
\(39\) 0 0
\(40\) 3.33428e7 − 5.77514e7i 0.0514840 − 0.0891729i
\(41\) −7.80704e8 −1.05239 −0.526193 − 0.850365i \(-0.676381\pi\)
−0.526193 + 0.850365i \(0.676381\pi\)
\(42\) 0 0
\(43\) 2.17607e7 0.0225734 0.0112867 − 0.999936i \(-0.496407\pi\)
0.0112867 + 0.999936i \(0.496407\pi\)
\(44\) −4.03263e7 + 6.98472e7i −0.0368637 + 0.0638498i
\(45\) 0 0
\(46\) 1.17650e8 + 2.03776e8i 0.0842219 + 0.145877i
\(47\) −1.51556e9 + 2.62504e9i −0.963910 + 1.66954i −0.251393 + 0.967885i \(0.580889\pi\)
−0.712517 + 0.701655i \(0.752445\pi\)
\(48\) 0 0
\(49\) 1.56382e9 − 1.21008e9i 0.790876 − 0.611976i
\(50\) −1.42997e9 −0.647131
\(51\) 0 0
\(52\) 4.03742e8 + 6.99301e8i 0.147260 + 0.255062i
\(53\) −1.98519e9 − 3.43845e9i −0.652057 − 1.12940i −0.982623 − 0.185612i \(-0.940573\pi\)
0.330566 − 0.943783i \(-0.392760\pi\)
\(54\) 0 0
\(55\) −1.60288e8 −0.0429443
\(56\) 1.37882e9 − 4.71168e8i 0.334559 − 0.114325i
\(57\) 0 0
\(58\) 1.58884e9 − 2.75195e9i 0.317853 − 0.550537i
\(59\) 1.37766e9 + 2.38619e9i 0.250875 + 0.434528i 0.963767 − 0.266746i \(-0.0859483\pi\)
−0.712892 + 0.701274i \(0.752615\pi\)
\(60\) 0 0
\(61\) −5.25879e9 + 9.10849e9i −0.797208 + 1.38080i 0.124220 + 0.992255i \(0.460357\pi\)
−0.921428 + 0.388550i \(0.872976\pi\)
\(62\) −6.45867e9 −0.895342
\(63\) 0 0
\(64\) 1.07374e9 0.125000
\(65\) −8.02391e8 + 1.38978e9i −0.0857753 + 0.148567i
\(66\) 0 0
\(67\) 1.14527e9 + 1.98367e9i 0.103633 + 0.179497i 0.913179 − 0.407559i \(-0.133620\pi\)
−0.809546 + 0.587056i \(0.800287\pi\)
\(68\) −5.23908e9 + 9.07435e9i −0.436974 + 0.756862i
\(69\) 0 0
\(70\) 2.18106e9 + 1.90492e9i 0.155106 + 0.135469i
\(71\) −4.54366e9 −0.298872 −0.149436 − 0.988771i \(-0.547746\pi\)
−0.149436 + 0.988771i \(0.547746\pi\)
\(72\) 0 0
\(73\) −6.98203e9 − 1.20932e10i −0.394191 − 0.682758i 0.598807 − 0.800893i \(-0.295642\pi\)
−0.992997 + 0.118135i \(0.962308\pi\)
\(74\) 6.25278e9 + 1.08301e10i 0.327566 + 0.567361i
\(75\) 0 0
\(76\) −2.11562e10 −0.957113
\(77\) −2.63788e9 − 2.30390e9i −0.111059 − 0.0969985i
\(78\) 0 0
\(79\) −1.39090e10 + 2.40910e10i −0.508564 + 0.880859i 0.491387 + 0.870941i \(0.336490\pi\)
−0.999951 + 0.00991713i \(0.996843\pi\)
\(80\) 1.06697e9 + 1.84805e9i 0.0364047 + 0.0630547i
\(81\) 0 0
\(82\) 1.24913e10 − 2.16355e10i 0.372075 − 0.644452i
\(83\) 6.17383e10 1.72038 0.860192 − 0.509971i \(-0.170344\pi\)
0.860192 + 0.509971i \(0.170344\pi\)
\(84\) 0 0
\(85\) −2.08241e10 −0.509053
\(86\) −3.48172e8 + 6.03052e8i −0.00798091 + 0.0138233i
\(87\) 0 0
\(88\) −1.29044e9 − 2.23511e9i −0.0260666 − 0.0451486i
\(89\) 1.20759e10 − 2.09160e10i 0.229231 − 0.397040i −0.728350 − 0.685206i \(-0.759712\pi\)
0.957580 + 0.288166i \(0.0930456\pi\)
\(90\) 0 0
\(91\) −3.31811e10 + 1.13386e10i −0.557395 + 0.190472i
\(92\) −7.52962e9 −0.119108
\(93\) 0 0
\(94\) −4.84981e10 − 8.40011e10i −0.681587 − 1.18054i
\(95\) −2.10228e10 − 3.64125e10i −0.278747 − 0.482804i
\(96\) 0 0
\(97\) −1.03350e11 −1.22198 −0.610991 − 0.791637i \(-0.709229\pi\)
−0.610991 + 0.791637i \(0.709229\pi\)
\(98\) 8.51348e9 + 6.26991e10i 0.0951401 + 0.700677i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 126.12.g.d.37.2 8
3.2 odd 2 42.12.e.d.37.3 yes 8
7.4 even 3 inner 126.12.g.d.109.2 8
21.11 odd 6 42.12.e.d.25.3 ✓ 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.12.e.d.25.3 ✓ 8 21.11 odd 6
42.12.e.d.37.3 yes 8 3.2 odd 2
126.12.g.d.37.2 8 1.1 even 1 trivial
126.12.g.d.109.2 8 7.4 even 3 inner